The Stacks project

Proof. Let $R$ be quasi-excellent. Using that a finite type algebra over $R$ is quasi-excellent (Lemma 15.52.2) we see that it suffices to show that any quasi-excellent domain is N-1, see Algebra, Lemma 10.162.3. Applying Algebra, Lemma 10.161.15 (and using that a quasi-excellent ring is J-2) we reduce to showing that a quasi-excellent local domain $R$ is N-1. As $R \to R^\wedge $ is regular we see that $R^\wedge $ is reduced by Lemma 15.42.1. In other words, $R$ is analytically unramified. Hence $R$ is N-1 by Algebra, Lemma 10.162.10. $\square$